Devil May Cry 5 has some very in-your-face story development, there's plenty of it in the game. Especially if you've played the previous titles. So I'm not really sure where you're coming from there. I would even go as far as to say it has one of the more enjoyable - possibly even one of the better - stories I've played this generation because instead of trying super hard to make an emotional pretentious movie-like experience, they somehow were able to make really cheesy dialogue work in both comedic and dramatic settings with a large cast of endearing characters. That being said, you probably won't enjoy the story if you haven't played the previous games, but I don't think that alone would erase the existence of story development from the game.
As for the puzzles and platforming, you're right that that kind of stuff is almost entirely absent (there are no puzzles but there is some platforming). But this in my opinion is actually a positive - it was a decision made by the Director because of fan feedback. Puzzles and platforming took away from the purest form of fun in an action game - the action. It was a way to remove filler.
In that regard, Devil May Cry 3 is the most "normal" of the Devil May Cry games. It almost feels like a precursor to the God of War format in some ways. There are a lot of "puzzles" and actions you do outside of combat. You also do a lot of traveling around inter-connected areas. Each boss feels sort of self-contained in a way, like they are their own story that you take part in, with an intro skit and an outro skit (which is in DMC5 as well but not to the same degree). It really helps to make you feel like you're part of a grander narrative, but it wasn't very substantive and actually detracted from the game in many ways. It's funny how elements like this might help the atmosphere and catharsis of the overall package - while detracting from the moment to moment enjoyment. I think the removement of these elements was for the better, but that's just my opinion.
Devil May Cry certainly isn't for everyone and it's fine to enjoy different things.